87962 - Statistical Field Theory

Learning outcomes

At the end of the course the student will learn the foundations of the physics of phase transitions and critical phenomena, within a framework common to Statistical Mechanics and Quantum Field Theory. He/she will be able to understand the physics of systems with an infinite number of degrees of freedom non-perturbatively through the methods of the renormalization group. The student will also be able to discuss and solve related physical problems.

Course contents

Here we give a detailed syllabus of the topics covered in both modules of the course, with references to the relevant chapters/sections of the recommended textbooks for study (see below in Readings/Bibliography).

Part 1 (module 2, Prof. L. Piroli)

Review of Classical and Quantum Statistical Mechanics

• Postulate of equal a priori probabilities.

• Microcanonical ensemble for classical systems of particles

• Definition of entropy and temperature.

• Canonical ensemble.

• Review of quantum statistical mechanics

See standard books on statistical mechanics, e.g. Ref. [1]

Phase transitions

• Definition and classification of phase transitions.

• Examples: condensation of a gas into a liquid and paramagnetic-ferromagnetic transition in a ferromagnet.

• Critical point and order parameter.

• Introduction to critical exponents.

Ch. 1.3, 1.4 in Ref. [2] and Ch. 1 in Ref. [3].

Mean-field theory

• Introduction to classical Ising model.

• Mean-field solution and mean-field free energy.

• Spontaneous symmetry breaking and emergence of an order parameter.

• Derivation of critical exponents in the mean-field approximation.

Ch. 5 in Ref [4], Ch. 2 in Ref. [3].

Exact analysis of the Ising model

• Exact solution of the Ising model in 1D via the transfer matrix approach.

• Computation of the correlation functions and the correlation length.

• Peierls’s argument for the existence of ferromagnetic order in 2D.

• Statement of the Kramers-Wannier duality and the exact result by Onsager.

Ch. 6.2 in Ref. [2]. Ch. 14.3 in Ref. [1].

The Landau-Ginzburg theory

• From lattice partition functions to field path integrals.

• Derivation of the Landau-Ginzburg free energy close to the critical point based on physical principles.

• Saddle-point approximation and definition of lower and upper critical dimension.

• Saddle-point solution with non-uniform boundary conditions: domain-walls.

Ch. 2.1, 2.2, 2.3, 2.4 in Ref. [2]. Sec. 1 in Ref. [5].

Fluctuations

• Computation of quadratic corrections close to the critical points and derivation of the upper critical dimension for the Ising model.

• Correlation functions and Ginzburg criterion.

Selected parts from Ch. 3 in Ref. [2] and from Sec. 2 in Ref. [5].

The scaling hypothesis

• (Generalized) homogeneous functions.

• Scaling hypothesis and derivation of the exponent identities.

• Hyperscaling relations.

Ch. 4.1, 4.2, 4.3 in Ref. [2]

Renormalization group (RG)

• Exact solution to the 1D Ising model via real-space RG.

• Conceptual general introduction within the Landau-Ginzburg framework.

• Linearization of the RG flow close to the critical point.

• Scaling directions, anomalous dimensions and classification of relevant, irrelevant, and marginal operators.

• Derivation of the scaling hypothesis from the RG framework

Ch. 6.3 in Ref. [2] (exact solution of the 1D Ising model). Ch. 4.4, 4.5 in Ref.[2]. Ch. 3 in Ref. [3]. Selected parts from Sec. 3 in Ref. [5].

The Gaussian model

• Exact solution of the Gaussian model and exact analysis via the momentum-space RG approach.

• Relevance of perturbation terms depending on the dimension.

• Definition of beta functions and remark about dangerously irrelevant operators.

Ch. 4.5 and 4.6 in Ref. [2]. Selected parts from Sec. 3 in Ref. [5].

Part 2 (module 1, Prof. F. Ravanini)

Recall of some Quantum Field Theory concepts

• Path integrals in QM and in QFT.

• Correlation Functions, Wick rotation and Euclidean formalism, Wick theorem.

• Symmetries and conservation laws, Nöther theorem, Ward identities.

• Energy-momentum tensor.

Ch. 2 (Sect. 2.1 to 2.5) in Ref. [7]

Conformal Field Theory in D dimensions

• Link between Quantum Field Theory and Statistical Mechanics.

• Algebra of local fields.

• Conformal transformations in D dimensions.

• Polyakov theorem

• Quasi-primary fields

Sect.. 10.1 to 10.4 in Ref. [6] and Ch. 4 in Ref. [7]

Conformal invariance in D=2

• 2D Conformal transformations.

• Classical conformal (de Witt) algebra in D=2.

• Ward identities and primary fields.

• Free massless bosons and fermions.

• Central charge and Virasoro algebra.

• Hamiltonian on a cylinder and Casimir effect.

• Representation theory: space of conformal states and space of conformal fields.

• Operator product expansions (OPE).

• Conformal bootstrap.

Sect. 10.5 to 10.9 in Ref. [6] and Ch. 5 and 6 in Ref. [7]

Minimal models

• Verma moduli, null vectors and degenerate representations.

• Gram matrix and Kac determinant.

• Minimal models, Kac tables.

• Counting of states, Theta functions and Virasoro characters.

• Differential equations for correlation functions.

• OPE and fusion rules.

• Examples of universality classes in D=2 for minimal models.

• Feigin-Fuchs construction and Coulomb gas formulation.

• Landau-Ginzburg and minimal models.

Sect. 11.1 to 11.6 in Ref. [6] and Ch. 8-9 in Ref. [7]

Modular invariance

• Modular group.

• Modular invariant partition function.

• Partition functions of minimal models.

• Orbifolds and exceptional theories.

• ADE classification of minimal models.

Sect. 11.7 in Ref. [6] and Ch. 10 in Ref. [7]

Free boson and free fermion theories

• c=1 CFT's and their classification.

• Free fermions: Neveu-Schwarz and Ramond sectors.

• Free fermions and Ising model.

Sect. 12.1 to 12.3 in Ref. [6]

Readings/Bibliography

Lecture Notes and Slides:

The material corresponding to the lectures, in the form of notes or slides used by the instructors, is available on the course’s Virtuale webpage.

Some exercises, both as assigned homework and self-assessment quizzes, are also available on the same Virtuale platform.

 

Recommended Textbooks:

Below is the list of reference texts indicated at the end of each section of the previous syllabus and strongly recommended for the study of the various topics.

The web links point to the PDF version of the book, if available online. In most cases, they are accessible through the SBA-Unibo online library service, which is available to regularly enrolled students at the University of Bologna.

1. K. Huang, Statistical Mechanics, John Wiley & Sons, New York
2. M. Kardar, Statistical Physics of Fields [https://research-ebsco-com.ezproxy.unibo.it/c/cavj5t/search/details/6wjmtue5br?db=e000xww], Cambridge University Press, 2007
3. J. Cardy, Scaling and Renormalization in Statistical Physics [https://www-cambridge-org.ezproxy.unibo.it/core/books/scaling-and-renormalization-in-statistical-physics/924C0B0D39123F681CF3353C42E5E836], Cambridge University Press, 1996
4. D. Tong, Lectures on Statistical Physics [https://www.damtp.cam.ac.uk/user/tong/statphys.html], available online at DAMTP, Cambridge, UK
5. D. Tong, Lectures on Statistical Field Theory [https://www.damtp.cam.ac.uk/user/tong/sft.html], available online at DAMTP, Cambridge, UK
6. G. Mussardo, Statistical Field Theory [https://academic-oup-com.ezproxy.unibo.it/book/43943?login=true&token=eyJhbGciOiJub25lIn0.eyJleHAiOjE3NTU1MzE1NzYsImp0aSI6Ijc2OGY3M2U0LTFhNmQtNDM2OS1iZWMwLTljODZlM2NjYzBjNyJ9.], Oxford Univ. Press
7. P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory [https://ebookcentral-proquest-com.ezproxy.unibo.it/lib/unibo/detail.action?docID=3075784], Springer, Berlin
   

Additional texts are also recommended for further study of the topics covered or for deeper insight into advanced developments:

• On lattice models and their exact solutions the ultimate book to consider is:
  
  R. Baxter, Exactly Solved Models in Statistical Mechanics [https://physics.anu.edu.au/research/ftp/_files/Exactly.pdf], Academic Press, London

• Many original articles in the early developments of Conformal Field Theory are available in the reprint book:
  
  C. Itzykson, H. Saleur, J.-B. Zuber, Conformal Invariance And Applications To Statistical Mechanics [https://research-ebsco-com.ezproxy.unibo.it/c/cavj5t/search/details/mlg65kdfef?db=e000xww], World Scientific, Singapore

• This review of 1988 written for the Les Houches summer school is one of the first and best places to find a full treatment of 2D CFT other than the original articles:
  
  P. Ginsparg, Applied Conformal Field Theory, Les Houches lectures 1988 – arXiv:hep-th/9108028 

• Another book where many advanced arguments of Statistical Field Theory can be found and a very mathematically rigorous introduction to CFT is present in Ch. 9 (Vol. 2) is:
  
  C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Cambridge University Press: Vol.1 and Vol.2

Teaching methods

Theoretical topics are fully explained in class by the teacher.
Some classes will be devoted to exercises that students will solve under the teacher's supervision.

Further exercises will be proposed on the Virtuale site as personal training.

Assessment methods

Oral exam at the blackboard: 2 or 3 questions chosen by the teacher on topics covered in class.

The purpose of the oral exam is to assess the student’s understanding of the concepts presented in the course, their ability to apply them to problem-solving, and their capacity to carry out the necessary logical and deductive reasoning.

Evaluation criteria will take into account:

• Balance between clarity and concision
• Mastery of the subject
• Lucidity of the presentation

The final grade will be expressed out of thirty, corresponding to the following performance evaluation parameters:

30 with honors (30 e lode): awarded only in exceptional cases, for an outstanding performance demonstrating complete mastery and creativity in the responses.

30: excellent performance with full command of the subject.

28–29: very good performance, demonstrating solid understanding of the concepts with minimal hesitation.

26–27: good performance, with some uncertainty or hesitation, but overall convincing.

22–25: satisfactory performance, with various uncertainties, though not on essential points.

18–21: weak performance, showing significant gaps, but still acceptable as a basic understanding of the core concepts.

Below 18 (Fail): insufficient performance, lacking knowledge of key topics and essential concepts of the course.

Students with Specific Learning Disabilities (SLD) or temporary/permanent disabilities are advised to contact the University Office responsible in a timely manner (https://site.unibo.it/studenti-con-disabilita-e-dsa/en). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, and will be evaluated in light of the learning objectives of the course.

 

Teaching tools

The lectures are presented mainly with slides, complemented by explanations at the blackboard.

A few exercises will be proposed in some of the subjects treated, by using the tools present on the Virtuale web page.

Office hours

See the website of Francesco Ravanini

See the website of Lorenzo Piroli